Show that the vector space $\text{Bil}(V)$ of all bilinear forms on $V$ can be decomposed in to the direct sum of $\text{Bil}(V)_{\text{sym}} \bigoplus \text{Bil}(V)_{\text{alt}}$, where $\text{Bil}(V)_{\text{sym}} $ and $\text{Bil}(V)_{\text{alt}}$ represent the subspaces of symmetric and alternating bilinear forms on $V$ respectively.
By a given hint I simply assume that they're vector spaces so I don't have to prove that but I have a hard time proving results regarding the direct sum of vector spaces in general. So far I have proven two other results prior to this question namely:A bilinear form $\mathfrak{b}$ is alternating if $\mathfrak{b}(v,w)=-\mathfrak{b}(w,v)$ holds true and $\mathfrak{b}$ is alternating when the Gram matrix is skew-symmetric.
Intuitively the statement makes sense to me, since we define symmetric bilinear forms as $\mathfrak{b}(v,w)=\mathfrak{b}(w,v)$ and alternating forms can be expressed as $\mathfrak{b}(v,w)=-\mathfrak{b}(w,v)$ but I don't know how I can go about proving it. I'd appreciate the help.